In a small bowl, combine eggs, milk, salt, basil and pepper. Melt remaining butter in skillet until foamy, swirling it around pan to coat evenly. When hot enough that a drop of water sizzles when dropped in, pour in egg mixture. Tip pan so eggs coat skillet evenly. As eggs cook, periodically lift up cooked edges, tilt pan and let uncooked egg run underneath.

When eggs are cooked, but surface is still shiny, place asparagus and mushrooms on one side; slide out of pan, folding side without vegetables over top.